Math 412 (Spring 2003) 


Instructor: Ed Bueler

Geometry is most interesting when it varies from place to place.  Gauss discovered, as a surveyor, the connection between the angles in a triangulation of the surface of the earth and the amount by which maps of the earth cannot be flattened without deforming or tearing.  Riemann took this understanding and extracted the fundamental definitions.  Then he generalized to any dimension.  After discovering special relativity, which is a new but flat geometry, Einstein realized how to slightly change the Riemannian tools to apply to gravity--he discovered that gravity is the amount by which the geometry of our space varies from place to place.  For the last century, mathematicians and physicists have been trying to understand how quantum mechanics works when the laboratory is curved, which we know it is!

This course will introduce the mathematics for the story.  You need to know multivariable calculus and how do both calculations and proofs.  With those tools you will get a quite complete understanding of the geometry of curves and surfaces, and an introduction to the general theory of Riemannian geometry.

Differential geometry is basic to the fundamental understanding of:


MATH F412 Differential Geometry,     Section F01
CRN 38058  
Class Time: TTh 3:40--5:10 pm         Classroom: Chapman 107.

(MORE DETAILS):    Instructor contact info:   Office: Chapman 301C             Phone: 474-7693
eMail:                         Web Site:

Course Description:   Differential geometry of curves and surfaces leading toward an abstract view of spaces.  The first fourth of the course will be the geometry of curves, which was introduced in calculus III--we will do a more complete job.  The next half of the course will be the geometry of surfaces.  We will see the contributions of Gauss and Riemann and lay the foundations for the geometry of curved manifolds of any dimension.  The last fourth of the course will be a survey of, and introduction to, manifolds, Riemannian geometry, Maxwell's equations and gravitation.
Course Grading:   Letter grade based on 40% homework and 60% exams.
Textbook:  Do Carmo, Differential Geometry of Curves and Surfaces, 1975.